If there is room left in the cells with the seedlings, you can sprinkle some grit or very coarse sand, or perlite, or biggish bark shreds on top of the mix surface. Water will run rapidly through those with only a little water clinging. That way, the SURFACE will dry out faster than the soilless mix.
With much less fungus right on the surface, you might have fewer fungus gnats.
A fan might help a little, by keeping the surface drier and blowing some of the adults away.
Sticky yellow cards might also help.
But if you aim for an organic "soil" seedling mix where some nutrients are expected to come from compost breaking down IN the seedling mix, you will have lots of fungus in the mix.
I would expect to have fungus gnats if I was trying to create a living soil community in my seedling mix.
I think most people, or at least almost all commercial growers and large-volume seed starters, stick to a near-sterile soilless mix for starting seedlings. I think their first concern was "damping off", and clean mixes and clean pots or cells help prevent damping off.
Also, how long does a seedling spend in a seed-starting tray? A few weeks? I think many seedlings get "transplanted out or potted up" in 2-4 weeks. I'd be amazed if that was long enough for a balanced soil ecosystem to evolve in a 2" x 2" seedling cell.
Also, seedlings don't need ANY nutrients for the first 2-4 weeks. Why gunk up the seedling mix with organics when they don't need the nutrients yet, and organics attract every kind of mold, fungus and plant disease?
If you want the seedlings to have early exposure to root fungi, maybe inoculate your seedling's mix or water with mycorrhizae? Then make sure they are short on N and P, to encourage the roots to take up the MR.
Maybe if the mix had a lot of finished compost in it, that would "stay healthy" and just continue to decompose for the few weeks it takes most seeds to emerge and get a few true leaves.
It's much easier to maintain the complex soil community in real, outdoor soil, over many years, with all the necessary variety of microbial and insect species available live or as spores on dust, from the living world around it. In the absence of all that living variety, small containers of real soil indoors may not stay healthy for long.
One species might grow in excess in the absence of its natural predators and regulators. Some other species might disappear from some cells because there were not enough of them to resist fluctuations, and interact as they would have in nature.
But everyone has their own, different, gardening process, and there's always someone who can "make it work" despite everyone else thinking "that's the wrong way!"
